# Fun Wave Eqn, Seperable Solution, Wave Guide

1. Nov 23, 2011

### autobot.d

1. The problem statement, all variables and given/known data
Boundaries at x=0,a y=0,b
This is a waveguide. w = angular frequency and k is the wavenumber.
Have the seperable solution to the wave equation.

$\psi$ = X(x)Y(y)e$^{i(kz-wt)}$

Where w=c$\sqrt{k^{2}+\pi^{2}\left(\frac{n^{2}}{a^{2}}+\frac{m^{2}}{a^{2}}\right)}$

I just need help with figuring out how to get this to an ODE and I should be able to figure out the boundary conditions. Thanks.

2. Relevant equations

3. The attempt at a solution

$\psi$$_{xx}$+$\psi$$_{yy}$-$\frac{1}{c^{2}}$$\psi$$_{tt}$=o

Plugging in

X''Ye$^{i(kz-wt)}$ + Y''Xe$^{i(kz-wt)}$ - k$^{2}$XYe$^{i(kz-wt)}$ + $\frac{w^{2}}{c^{2}}$XYe$^{i(kz-wt)}$ = 0

Now dividing by XY and factoring out e$^{i(kz-wt)}$

e$^{i(kz-wt)}$$[ \frac{X''}{X}$ + $\frac{Y''}{Y}$ - k$^{2}$ + $\frac{w^{2}}{c^{2}}]$ = 0

Does k$^{2}$ = $\frac{w^{2}}{c^{2}}$ necessarily?

Do I set $\frac{X''}{X} = \alpha$ and $\frac{Y''}{Y} = \beta$
where $\alpha , \beta$ are constants and solve these ODE's. Doesn't seem to get me where I want to go though.

Any advice would be very awesome. Thanks!
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution